Risk and Return
Spring 2022
Chapter 10
Corporate Finance: The Core (4th Edition)
Berk/DeMarzo
§Risk and Return
§Common measures of risk and return – mean, variance,
standard deviation
§Measuring historical returns
§Historical trade-off between risk and return
§Common vs. Independent Risk
§Stock Portfolio Diversification
§Measuring Systematic Risk
§Beta – the sensitivity of a security’s return to the return of the
overall market
§Beta and the Cost of Capital – the Capital Asset Pricing Model
(CAPM)
§Historical returns on different investments
§In 2015, $100 invested at the end of 1925 (dividends and
interest reinvested) will have grown into
§S&P 500: $480,560
§Small Stocks: $4,637,913
§World Portfolio: $187,182
§Corporate Bonds: $23,147
§Treasury Bills: $2,043
Source: Chicago Center for Research in Security Prices, Standard and Poor’s, MSCI, and Global Financial Data.
§Why invest in anything else than small stocks?
§People hardly make 89-year investments
§Small stocks outperformed other assets on average – they
also experienced periods of large losses, most likely in
situations when people need money most (Great
Depression, Financial Crisis)
§On the other hand, Treasury bills enjoyed small but steady
gains each year
Source: Chicago Center for Research in Security Prices, Standard and Poor’s, MSCI, and Global Financial Data.
§Small Stocks had the highest performance, but also the
most variable returns
§Depending on the investment horizon, small stocks can do
worse than even Treasuries
§The goal in this chapter is to quantify the relationship
between risk and return
§Probability Distributions
§Summarizes information regarding risky investments –
assigns a probability PR to each possible return R
§Example: BFI stock currently trades for $100 per share.
You believe that in one year there is a 25% chance of the
share price being $140, 50% chance that it will be $110, and
25% chance that it will be $80. BFI pays no dividends
§Expected Return
§Weighted average of the possible returns (weights
correspond to probabilities)
[ ]
Expected Return == ´
åR
R
ER P R
[ ]
25%( 0.20) 50%(0.10) 25%(0.40) 10%=-+ + =
BFI
ER
§Var iance
§Expected squared deviation from the mean
§Standard Deviation
§Square root of variance
[ ]
( )
[ ]
( )
22
()
éù
=-=´-
ëû
åR
R
Var R E R E R P R E R
() () =SD R Var R
§Both variance and standard deviation are measures of risk
§For BFI example,
§We refer to the standard deviation of a return as its
volatility
§TXU stock has the following probability distribution. What
is its expected return and volatility (standard deviation)?
Probability Return
.25 8%
.55 10%
.20 12%
§Solution
§Expected Return
E[R] = 0.25*0.08 + 0.55*0.10 + 0.20*0.12 = 0.099, or 9.9%
§Standard Deviation
Var[R] = 0.25*(0.08-0.099)^2 + 0.55*(0.10-0.099)^2 +
0.20*(0.12-0.099)^2 = 0.000179
SD[R] = sqrt(0.000179) = 0.01338, or 1.338%
§Computing Historical Returns
§Realized Return – the actual return observed over a particular
time period
§Realized Annual Returns (from quarterly returns)
annual 1 2 3 4
1 (1 )(1 )(1 )(1 )+=+ + + +
QQ Q Q
RRRRR
§Both dividends and capital gains contribute to total
realized return
§The returns are risky (in some years returns are positive, in
others they are negative)
§We can compute realized returns for any investment or
portfolio
§Average Annual Returns – the average of realized returns
from years 1 to T
§From Table 10.2, the average return for the S&P 500 for the years
2002-2014 is
( )
12
1
11
=
=+++=
å
!
T
Tt
t
RRR R R
TT
1
( 0.221 0.287 0.109 0.109 0.158 0.055 0.370
13
0.265 0.151 0.021 0.160 0.324 0.137) 8.7%
R=-+++++-
++++++ =
§Variance and Volatility of Returns
§Figure 10.5 shows variability of returns differs for each investment
§To quantify this difference, we can estimate the standard deviation
of the probability distribution
§The volatility (standard deviation) is the square root of the variance
§Small stocks have had the most variable historical returns
§Treasury bills are the least volatile investment category
§Estimation Error: Using Past Returns to Predict the Future
§We can use a security’s historical average return to infer its
expected return
§However, the average return is just an estimate of the true
expected return, and is subject to estimation error
§Estimation Error: Using Past Returns to Predict the Future
§Standard Error – estimation error of a statistical estimate
§95% confidence interval
§For the S&P during 1926-2014,
(Individual Risk)
(Average of Independent, Identical Risks) Number of Observations
=SD
SD
20.1%
12.0% 2 12.0% 4.3%
89
æö
±=±
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§The Returns of Large Portfolios
§Investments with higher volatility have rewarded investors
with higher average returns
§Riskier investments must offer investors higher average
returns to compensate them for the extra risk they are taking
on
§The Returns of Individual Stocks
§The risk-return trade-off we have seen for large portfolios do
not necessarily hold for individual stocks
§Larger stocks have lower volatility overall
§Even the largest stocks are typically more volatile than the
S&P 500
§There is no clear relationship between volatility and return
§Theft vs. Earthquake Insurance
§Each year, there is about a 1% chance of home robbery and
1% chance that the home will be damaged by an earthquake
§If an insurance company writes 100,000 policies of each type,
are the risks of the portfolios similar?
§Theft vs. Earthquake Insurance
§The risk of the earthquake insurance portfolio is no different from
the risk of any single policy – it is still all or nothing (common risk)
§For theft insurance, the number of claims in a given year is quite
predictable – the portfolio has almost no risk (independent risk)
§Common risk – risk perfectly correlated among securities
§Independent risk – risk uncorrelated and independent
across securities
§Diversification – averaging out of independent risks in a
large portfolio
§The Role of Diversification
§Standard deviation for the homeowner for each type of insurance
§For the earthquake insurer, the risk is common, so the percentage
of claims received is also 1% on average, with 9.95% SD
§The Role of Diversification
§Standard deviation for the theft insurer
§There is almost no risk for the theft insurer
§Independent risks are diversified in a large portfolio,
whereas common risks are not
§Firm-Specific vs. Systematic Risk
§Firm-specific news is good or bad news about an individual
company
§Market-wide news is news about the economy as a whole,
affecting all stocks
§Firm-specific risk
§Idiosyncratic, Unique, Diversifiable risk
§Systematic risk
§Undiversifiable, Market risk
§Firm-Specific vs. Systematic Risk
§When many stocks are combined in a large portfolio, the firm-
specific risks for each stock will average out and be
diversified
§Systematic risk will affect all firms (and therefore the entire
portfolio) and will not be diversified
§No Arbitrage and the Risk Premium
§If independent risk can be diversified away, investors will not
be compensated for holding firm-specific risk
§Otherwise, holding a large portfolio will provide arbitrage
opportunity
§The risk premium of a security is determined by its systematic
risk and does not depend on its diversifiable risk
§No Arbitrage and the Risk Premium
§This implies that a stock’s volatility, which is a measure of total
risk, is not very useful in determining the risk premium
§To estimate a security’s expected return, we need to find a
measure of a security’s systematic risk (how the returns move
in relation to the overall economy)
§Identifying Systematic Risk: The Market Portfolio
§To determine how sensitive a stock is to systematic risk, we can look at
the average change in its return for each 1% change in the return of a
portfolio that fluctuates solely due to systematic risk (efficient portfolio)
§The market portfolio, which contains all stocks and securities, is a natural
candidate of the efficient portfolio
§In practice, the S&P 500 portfolio is used, assuming it is fully diversified